Synchronization between devices

ABSTRACT

The present disclosure relates to a method to determine a clock signal when separate clocks are used. In one embodiment, a disciplined clock system comprising an update subsystem and a synthesis subsystem is provided. A first clock phase estimate is provided to the update subsystem and used, along with the update subsystem, to determine a frequency offset estimate and a phase offset estimate. The clock signal is determining using the frequency offset estimate, the phase offset estimate, and the synthesis subsystem. Alternatively, two clocks can be synchronized by generating a signal associated with a first clock; modulating the signal; transmitting the modulated signal; receiving the modulated signal by a receiver associated with a second clock; correlating the received signal; determining the time of arrival of the received signal; determining the time difference between the two clocks; and synchronizing the two clocks.

CROSS-REFERENCE TO OTHER APPLICATIONS

This application is a divisional application of U.S. patent applicationSer. No. 12/817,355, filed on Jun. 17, 2010, which is acontinuation-in-part of international application numberPCT/US2010/032606, filed Apr. 27, 2010, which claims the benefit of U.S.Provisional Application No. 61/173,382, filed Apr. 28, 2009.

BACKGROUND

1. Technical Field

The present disclosure relates generally to the logging of subsurfaceformations surrounding a wellbore using a downhole logging tool, andparticularly to making measurements with a modular logging tool whiledrilling, and using those measurements to infer one or more formationproperties.

2. Background Art

Logging tools have long been used in wellbores to make, for example,formation evaluation measurements to infer properties of the formationssurrounding the borehole and the fluids in the formations. Commonlogging tools include electromagnetic tools, nuclear tools, and nuclearmagnetic resonance (NMR) tools, though various other tool types are alsoused.

Early logging tools were run into a wellbore on a wireline cable, afterthe wellbore had been drilled. Modern versions of such wireline toolsare still used extensively. However, the need for information whiledrilling the borehole gave rise to measurement-while-drilling (MWD)tools and logging-while-drilling (LWD) tools. MWD tools typicallyprovide drilling parameter information such as weight on the bit,torque, temperature, pressure, direction, and inclination. LWD toolstypically provide formation evaluation measurements such as resistivity,porosity, and NMR distributions (e.g., T1 and T2). MWD and LWD toolsoften have components common to wireline tools (e.g., transmitting andreceiving antennas), but MWD and LWD tools must be constructed to notonly endure but to operate in the harsh environment of drilling.

Electromagnetic (EM) wave propagation in a medium is characterized bythe magnetic permeability of the medium (μ) and the complex dielectricpermittivity (∈*) given by,

$\begin{matrix}{ɛ^{*\;} = {ɛ_{r} - {i\frac{\sigma}{{\varpi ɛ}_{0}}\mspace{14mu}{and}}}} & (1) \\{\mu = {\mu_{r}{\mu_{0}.}}} & (2)\end{matrix}$∈_(r) and μ_(r) are the permittivity and permeability of the mediumrelative to their corresponding values in free space (∈₀=8.8 10⁻¹², andμ₀=1/(4π10⁻⁷)), ω is the angular frequency, and σ is the conductivity.Those parameters affect the wave vector k, given by,

$\begin{matrix}{k = {\frac{\varpi}{c}\sqrt{\mu_{r}ɛ_{r}}}} & (3)\end{matrix}$where c, the speed of light in vacuum, is given by,

$\begin{matrix}{c = {\frac{1}{\sqrt{\mu_{0}ɛ_{0}}}.}} & (4)\end{matrix}$

Most rocks of interest are non-magnetic and therefore μ_(r) equals one.An EM measurement from a resistivity logging tool is related to k, whichin turn is related to ∈_(r) and σ. The real and imaginary parts of ∈*have different frequency dependencies. For example, the conductivity istypically constant until the frequency is above about 1 MHz, after whichit increases slowly. The permittivity of rocks, on the other hand isvery large (e.g., ˜10⁹) at sub-Hz frequencies, and decreases as thefrequency increases, but eventually flattens out at frequencies around aGHz. The frequency dependence of permittivity is 1/f for frequencies upto approximately 10⁴ Hz, but between 10⁴ and 10⁸ Hz, it varies as1/(f^(α)), where CC is approximately 0.3. Since the imaginary part of ∈*has an explicit 1/f dependence, the imaginary part dominates at lowfrequency and the real part dominates at high frequencies.

Most prior art low frequency resistivity tools have concentrated on theconductivity term of the complex permittivity and ignored the real part(which is known as the dielectric constant). As such, those tools onlymeasure the amplitude of the received signal, which is sufficient tosolve for the conductivity. However, if the phase of the received signalis also measured, one can additionally solve for the real and imaginarypart of the complex permittivity. There is increasingly more interest inthe dielectric constant since it contains information on themicro-geometry of the rock matrix.

Physics-based models explaining the frequency dependence of permittivity(and specifically the dielectric constant) attribute the variation withfrequency to three effects, each of which operates in a particularfrequency range. At high frequencies, where the permittivity isessentially frequency independent, the permittivity of the rock, whichis a mixture of the solid matrix, water, and hydrocarbons, can becalculated using the “complex refractive index method” (CRIM), shown byEquation (5) below,√{square root over (∈_(rock)*)}=(1−φ)√{square root over (∈_(matrix))}+S_(water)φ√{square root over (∈_(Water)*)}+(1−S _(water))φ√{square rootover (∈_(hydrocarbon))}.  (5)This is a simple volumetric average of the refractive index (that is,the square root of the permittivity) of the components. Any slightfrequency dependence in this range is the result of the frequencydependence of the water permittivity.

The intermediate range, where the permittivity varies as the (−α) powerof frequency, is attributed to the geometrical shape of the rock grains.The insulating grains, surrounded by conductive water, form localcapacitors that respond to the applied electric field. The permittivityin this range has been described by several models, one of which, for afully water-filled rock, is given by,

$\begin{matrix}{\phi = {\left( \frac{ɛ_{rock}^{*} - ɛ_{matrix}}{ɛ_{water}^{*} - ɛ_{matrix}} \right)\left( \frac{ɛ_{water}^{*}}{ɛ_{rock}^{*}} \right)^{L}}} & (6)\end{matrix}$where L is the depolarizing factor describing the average grain shape.For example, L is ⅓ for spherical grains, and it deviates for morerealistic, spheroidal grain shapes, though it remains between 0 and 1.This equation can be easily modified to include partial water saturationand the effect of hydrocarbons on the measured complex permittivity. Asmentioned above, the intermediate frequency range starts atapproximately 100 kHz, which is the operating frequency of mostpropagation and induction tools, so this expression is very applicableto the measurements from these tools and leads to a complex permittivityof water from which water salinity can be determined. The expressionalso provides a measure of grain shape that has further application.

At frequencies below 100 kHz, the permittivity has a 1/f dependence.This is attributed to the double layer effects caused by surface chargeson the surfaces of the rock grains. The surfaces of the rock grains arecharged either by the nature of the minerals at the surface, or, moreimportantly, by the varying amounts of clay mineral at the surface.These minerals have surface charges in contact with a cloud ofoppositely charged counter-ions, forming an ionic double layer. Thecounter-ions respond to the applied electric field and cause a largepermittivity. The permittivity in this frequency range is a clayindicator and can be used to estimate the clay concentration orcation-exchange capacity (CEC). Thus, any resistivity tool that measuresthe amplitude and phase of the received signal below 100 kHz candetermine the conductivity and permittivity of the rock and can providean estimate of the shale content. In addition to shale estimation, phasemeasurement may be used to determine a phase conductivity in addition tothe traditionally measured amplitude conductivity. It has been shownthat those two responses have different depths of investigations, andtheir combination provides a very good bed boundary indicator.

SUMMARY

The present disclosure relates to a method to determine the phase of asignal when transmitter and receiver circuits use separate clocks. Adiscrepancy between the separate clocks is determined, as is acorrection factor between the separate clocks. The phase is determinedusing a measured time of arrival of the signal, the determineddiscrepancy, and the determined correction factor. A drift factor and anexpected start time of a pulse sequence may be used to determine thediscrepancy. A start time of a pulse within the pulse sequence isdetermined and used to determine the correction factor. The method worksby either absolute synchronization of the separate clocks, or by makingthe measurements independent of clock synchronization.

The present disclosure further relates to a method to determine a clocksignal when separate clocks are used. In one embodiment, a disciplinedclock system comprising an update subsystem and a synthesis subsystem isprovided. A first clock phase estimate is provided to the updatesubsystem and used, along with the update subsystem, to determine afrequency offset estimate and a phase offset estimate. The clock signalis determining using the frequency offset estimate, the phase offsetestimate, and the synthesis subsystem. Alternatively, two clocks can besynchronized by generating a signal associated with a first clock;modulating the signal; transmitting the modulated signal; receiving themodulated signal by a receiver associated with a second clock;correlating the received signal; determining the time of arrival of thereceived signal; determining the time difference between the two clocks;and synchronizing the two clocks.

Other aspects and advantages will become apparent from the followingdescription and the attached claims.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates an exemplary well site system.

FIG. 2 shows a prior art electromagnetic logging tool.

FIG. 3 is a schematic drawing of a typical modular resistivity tool withexemplary transmitter and receiver spacings, in accordance with oneembodiment in the present disclosure.

FIG. 4 shows an exemplary sequence with three frequencies, differentpulse widths, and time between pulses, in accordance with one embodimentin the present disclosure.

FIG. 5 illustrates various specific times and time intervals and howthey relate to one another, in accordance with one embodiment in thepresent disclosure.

FIG. 6 illustrates how to synchronize the subs with the master clock, inaccordance with one embodiment in the present disclosure.

FIG. 7 shows a simulated received pulse after it has been transmittedthrough a formation, in accordance with one embodiment in the presentdisclosure.

FIG. 8 shows the result of coarse processing when the pulse of FIG. 7 ismatch filtered.

FIG. 9 shows a coarse processing flow beginning with an incoming signaland resulting in a phase independent amplitude output, in accordancewith one embodiment in the present disclosure.

FIG. 10 is a flowchart showing the steps taken in a specific embodiment,in accordance with the present disclosure.

FIG. 11 schematically represents a first implementation method of adevice, in accordance with the present disclosure.

FIG. 12 schematically represents an embodiment of a disciplined clocksystem, in accordance with the present disclosure.

FIGS. 13A and 13B are plots comparing a noisy clock, an ideal clock, anda slave clock, in accordance with the present disclosure.

FIG. 14 schematically represents a clock-time transfer performed byperiodically sending a timing signal from an MCD to an SCD, inaccordance with the present disclosure.

FIG. 15 schematically represents one embodiment of a synchronizedmeasurement system, in accordance with the present disclosure.

FIG. 16 is a plot illustrating a noisy clock in time and frequency, inaccordance with the present disclosure.

FIG. 17 schematically represents a functional diagram of one embodimentof a phase-locked loop, in accordance with the present disclosure.

FIG. 18 is a set of graphs showing various pulse shapes used in testing,in accordance with the present disclosure.

FIG. 19 is a plot of results using a super-resolution method fordifferent pulse shapes, in accordance with the present disclosure.

FIG. 20 is a plot showing phase and frequency error traces of the updatesubsystem for one embodiment, in accordance with the present disclosure.

FIG. 21 is a plot showing frequency estimate and frequency error of theupdate subsystem for one embodiment, in accordance with the presentdisclosure.

FIG. 22 is a plot of simulation results comparing an “old” method withthe “new” method, in accordance with the present disclosure.

FIG. 23 is a plot showing the performance of different methods andsignals versus the Cramer-Rao bound, in accordance with the presentdisclosure.

FIG. 24 is a plot comparing the frequency estimates from an “old” methodand a phase-locked loop method, in accordance with the presentdisclosure.

FIG. 25 is a flowchart showing steps in one embodiment, in accordancewith the present disclosure.

FIG. 26 is a plot showing correlation values for various phase offsets,in accordance with the present disclosure.

FIG. 27 is a flowchart showing steps in one embodiment, in accordancewith the present disclosure.

DETAILED DESCRIPTION

Some embodiments will now be described with reference to the figuresLike elements in the various figures will be referenced with likenumbers for consistency. In the following description, numerous detailsare set forth to provide an understanding of various embodiments and/orfeatures. However, it will be understood by those skilled in the artthat some embodiments may be practiced without many of these details andthat numerous variations or modifications from the described embodimentsare possible. As used here, the terms “above” and “below”, “up” and“down”, “upper” and “lower”, “upwardly” and “downwardly”, and other liketerms indicating relative positions above or below a given point orelement are used in this description to more clearly describe certainembodiments. However, when applied to equipment and methods for use inwells that are deviated or horizontal, such terms may refer to a left toright, right to left, or diagonal relationship as appropriate.

FIG. 1 illustrates a well site system in which various embodiments canbe employed. The well site can be onshore or offshore. In this exemplarysystem, a borehole 11 is formed in subsurface formations by rotarydrilling in a manner that is well known. Some embodiments can also usedirectional drilling, as will be described hereinafter.

A drill string 12 is suspended within the borehole 11 and has a bottomhole assembly 100 which includes a drill bit 105 at its lower end. Thesurface system includes platform and derrick assembly 10 positioned overthe borehole 11, the assembly 10 including a rotary table 16, kelly 17,hook 18 and rotary swivel 19. The drill string 12 is rotated by therotary table 16, energized by means not shown, which engages the kelly17 at the upper end of the drill string. The drill string 12 issuspended from a hook 18, attached to a traveling block (also notshown), through the kelly 17 and a rotary swivel 19 which permitsrotation of the drill string relative to the hook. As is well known, atop drive system could alternatively be used.

In the example of this embodiment, the surface system further includesdrilling fluid or mud 26 stored in a pit 27 formed at the well site. Apump 29 delivers the drilling fluid 26 to the interior of the drillstring 12 via a port in the swivel 19, causing the drilling fluid toflow downwardly through the drill string 12 as indicated by thedirectional arrow 8. The drilling fluid exits the drill string 12 viaports in the drill bit 105, and then circulates upwardly through theannulus region between the outside of the drill string and the wall ofthe borehole, as indicated by the directional arrows 9. In this wellknown manner, the drilling fluid lubricates the drill bit 105 andcarries formation cuttings up to the surface as it is returned to thepit 27 for recirculation.

The bottom hole assembly 100 of the illustrated embodiment includes alogging-while-drilling (LWD) module 120, a measuring-while-drilling(MWD) module 130, a roto-steerable system and motor, and drill bit 105.

The LWD module 120 is housed in a special type of drill collar, as isknown in the art, and can contain one or a plurality of known types oflogging tools. It will also be understood that more than one LWD and/orMWD module can be employed, e.g. as represented at 120A. (References,throughout, to a module at the position of 120 can alternatively mean amodule at the position of 120A as well.) The LWD module includescapabilities for measuring, processing, and storing information, as wellas for communicating with the surface equipment. In the presentembodiment, the LWD module includes a resistivity measuring device.

The MWD module 130 is also housed in a special type of drill collar, asis known in the art, and can contain one or more devices for measuringcharacteristics of the drill string and drill bit. The MWD tool furtherincludes an apparatus (not shown) for generating electrical power to thedownhole system. This may typically include a mud turbine generatorpowered by the flow of the drilling fluid, it being understood thatother power and/or battery systems may be employed. In the presentembodiment, the MWD module includes one or more of the following typesof measuring devices: a weight-on-bit measuring device, a torquemeasuring device, a vibration measuring device, a shock measuringdevice, a stick/slip measuring device, a direction measuring device, andan inclination measuring device.

An example of a tool which can be the LWD tool 120, or can be a part ofan LWD tool suite 120A of the system and method hereof, is the dualresistivity LWD tool disclosed in U.S. Pat. No. 4,899,112 and entitled“Well Logging Apparatus And Method For Determining formation ResistivityAt A Shallow And A Deep Depth,” incorporated herein by reference. Asseen in FIG. 2, upper and lower transmitting antennas, T₁ and T₂, haveupper and lower receiving antennas, R₁ and R₂, therebetween. Theantennas are formed in recesses in a modified drill collar and mountedin insulating material. The phase shift of electromagnetic energy asbetween the receivers provides an indication of formation resistivity ata relatively shallow depth of investigation, and the attenuation ofelectromagnetic energy as between the receivers provides an indicationof formation resistivity at a relatively deep depth of investigation.The above-referenced U.S. Pat. No. 4,899,112 can be referred to forfurther details. In operation, attenuation-representative signals andphase-representative signals are coupled to a processor, an output ofwhich is coupleable to a telemetry circuit.

Recent electromagnetic logging tools use one or more tilted ortransverse antennas, with or without axial antennas. Those antennas maybe transmitters or receivers. A tilted antenna is one whose dipolemoment is neither parallel nor perpendicular to the longitudinal axis ofthe tool. A transverse antenna is one whose dipole moment isperpendicular to the longitudinal axis of the tool, and an axial antennais one whose dipole moment is parallel to the longitudinal axis of thetool. Two antennas are said to have equal angles if their dipole momentvectors intersect the tool's longitudinal axis at the same angle. Forexample, two tilted antennas have the same tilt angle if their dipolemoment vectors, having their tails conceptually fixed to a point on thetool's longitudinal axis, lie on the surface of a right circular conecentered on the tool's longitudinal axis and having its vertex at thatreference point. Transverse antennas obviously have equal angles of 90degrees, and that is true regardless of their azimuthal orientationsrelative to the tool.

The phase of a received signal wherein the transmitter and receivercircuits use their own separate clocks can be measured either byabsolute synchronization of the two clocks, or by making themeasurements independent of the clock synchronization.

Conventional resistivity tools use a single clock in the tool to sampleand record the time when the transmitter antenna is energized and whenthe associated wave is received by the receiver antenna. Since the timeinterval between the transmitting and receiving event is the parameterof interest, the clock does not have to be synchronized with any otherclock. The measurement is a time difference measurement and, as such, solong as the clock does not drift during the time period when these twoevents take place, there is no error due to clock discrepancy.

Modular resistivity tools have been proposed wherein different antennasare located in different modules. These modules can be placed atdifferent places within a bottomhole assembly (BHA), creating a desiredtransmitter-receiver (T-R) spacing and radial depth of investigation.Other LWD or MWD tools can occupy the space between the modules so thatthe space is not wasted. FIG. 3 shows an exemplary BHA containing amodular resistivity tool.

Specifically, FIG. 3 shows an arrangement of three modules in a BHA. TheBHA contains a drill bit, followed by a rotary steerable sub that makesdirectional drilling possible. In this example, the first sub (PERISCOPEtool) is located immediately above the rotary steerable sub, but ingeneral the locations of different modules can be different from one BHAto another. To create space between the first module and the second(Transmitter #2), an MWD sub has been used. The MWD sub serves as aspacer but also performs its own functions. If the desired distancebetween the modules can not be filled with an existing LWD tool,sections of drill pipe with no particular functionality can be used toachieve the desired distance. The length of BHA between the secondmodule and the third module (Transmitter #1) is filled using an LWDsonic sub in this example. Thus, the resistivity tool in this example ismade up of three modules separated by two LWD tools. The tool is locatedabove the rotary steerable sub as shown in FIG. 3. With separate modulesit is possible to choose different T-R spacing and thus different radialdepths of investigations.

Since each module preferably has its own clock, the modular tool designintroduces the clock synchronization and drift problems. The transmitter(TX) module and receiver (RX) module work independently, and the time ofthe transmitting and receiving events is usually measured by twodifferent clocks. Although these clocks can be set to be equal beforesending the tool into the borehole, the clocks have intrinsicallydifferent rates, and if left alone, the clocks will not measure the sametime at a later point. In addition, downhole temperature changes canaffect these clocks differently, and, since certain modules may beseparated by, say, 100 ft from other modules, the clocks in the separatemodules may be exposed to different local temperatures, causing anothersource of error.

Consider two different electronic circuits, one using a clock that weshall call “master”, M, and the other using a different clock called“local”, L. “M” or “L” may also be used herein to indicate or includethe respective circuitry associated with the master clock, M, or localclock, L. In general, there may be multiple modules operating, andsynchronization is generally needed between all modules involved in ameasurement. In the embodiments described below, only two modules areused, but the method is easily extendable to as many modules as desired.

For simplicity, assume the clock in the receiver circuit is the master,though it does not matter whether the RX or TX clock is designated asthe master clock. Further consider a sequence of transmission eventsfrom the transmitter antenna. This sequence may be a preset list ofpulses transmitted by the TX antenna into the formation, the sequencemay have pulses of different frequency and duration, and the timebetween pulses may be different. An exemplary sequence for an EM loggingtool is shown in FIG. 4. The transmitter may be programmed to transmit asequence such as one shown in FIG. 4 in a repetitious fashion.

One implementation to correct the clock drift works by M sending asynchronizing pulse to start an initiating cycle. This signal istypically sent along a conductive pathway through the BHA that serves asa communication link, often referred to as a “bus”. As the bus lengthincreases, for example, with increased module spacing, the variation inthe time of propagation of the synchronizing signal may becomenon-negligible. This variation in propagation time must be taken intoaccount to make accurate phase measurements.

An alternative way to send the synchronizing signal is as a transmittedwave passing through the formation. This involves, for example,transmitting various signals at a given frequency for some desired time.Alternatively, some of the pulses used for sampling the formation can beused as synchronization pulses. However, the formation signalpropagation time can vary and that variation must be accounted for orminimized. The effect of the formation on the propagation time istypically less than 100 ns, and if the separation betweensynchronization pulses is chosen to be on the order of seconds, theeffect of the formation variation will be negligible. Thus, the clockfrequency correction estimate can be made arbitrarily better byincreasing the time between observed synchronization signals. Theparticular amount of time depends on the acceptable tolerance. A localclock measures the time difference between the transmitted signals anduses a priori knowledge of the timing between those transmissions (i.e.,transmission sequence timing) to compute a correction factor that isaccurate to within the propagation uncertainty over the interval betweenthe received signals. Uncertainty in the formation signal propagationtime is due in part to changes along the wellbore (i.e., measured depth)and changes around the wellbore (i.e., tool rotation angle). Theuncertainty caused by rotation can be reduced by limiting considerationto only those signals that are acquired at the same, or nearly the same,tool rotation angle. Comparing the elapsed time between like-positionedsamples measured by the local clock with the interval time between thesamples according to the transmission sequence timing allows a clockfrequency correction to be computed.

For the present invention, either method of sending the synchronizingpulse or signal may be used. The time between two adjacentsynchronization pulses as measured by M, for example, is (T^(M)_(i)−T^(M) _(i−1)) and can be as short as the duration of one sequence,but typically is longer than the duration of a few sequences (see FIG.5). Note that we use a superscript to specify which clock has made themeasurement. The synchronizing pulses are measured by both the M and Lclocks. In each case the measurement is made by the clock in themeasuring circuit. Once M measures T^(M) _(i), it broadcasts that valueand L receives that information. L proceeds to calculate ΔM^(M)=T^(M)_(i)−T^(M) _(i−1) using the information provided, and also calculatesΔL^(L) from its own measurements of the same two pulses. If one clock isfaster than the other, ΔM^(M) and ΔL^(L) will be different and thecorrection factor, K_(ppb), will be different from zero,

$\begin{matrix}{K_{ppb} = {1 - {\frac{\Delta\; L^{L}}{\Delta\; M^{M}}.}}} & (7)\end{matrix}$

Note that K_(ppb) depends on the time difference between synchronizingpulses and is independent of the absolute time reading of either clock.Also, K_(ppb) is greater than zero if L is slower than M. L alsocalculates the expected times for starting the next sequence by the twoclocks, namely T_(si) ^(L) and T_(si) ^(M), as shown schematically inFIG. 5. Those values relative to the synchronization pulse will be,T _(SLi) ^(L) =T _(Si) ^(L) −T _(i) ^(L) and  (8)T _(SMi) ^(M) =T _(Si) ^(M) −T _(i) ^(M).  (9)

More detailed description of how those quantities are calculated isdescribed below. For simplicity, clock drifts are assumed to be close toconstant between synchronization pulses, which allows for easyderivations. Nonetheless, if necessary, more advance filtering can beimplemented through the use of past time differences of synchronizationpulses. In the next step, the start sequence time measured by L iscalculated with respect to M,

$\begin{matrix}{T_{SLi}^{M} = {\frac{T_{SLi}^{L}}{1 - K_{ppb}}.}} & (10)\end{matrix}$Having start sequence times for both clocks relative to M, they can besubtracted to calculate the discrepancy between the two clocks relativeto M,Δ_(i) ^(M) =T _(SMi) ^(M) −T _(Sli) ^(M).  (11)This error accounts for the lack of synchronization between the twoclocks and L sends this value to M. At this point, M can use K_(ppb) ascalculated before, or preferably, M may interpolate an updated value forK_(ppb) using information from the previous cycle,

$\begin{matrix}{K_{ppb} = {- {\frac{\left( {\Delta_{i}^{M} - \Delta_{i - 1}^{M}} \right)}{\left( {T_{Si}^{M} - T_{{Si} - 1}^{M}} \right) - \left( {\Delta_{i}^{M} - \Delta_{i - 1}^{M}} \right)}.}}} & (12)\end{matrix}$

The starting time of one of the pulses within the sequence relative tothe start time of the sequence is T_(f). This parameter, if measured byL and transformed in reference to M, is,

$\begin{matrix}{\left( T_{f}^{L} \right)^{M} = {\frac{T_{f}^{L}}{1 - K_{ppb}}.}} & (13)\end{matrix}$The difference as calculated below is the correction for a pulse withinthe sequence,ΔT _(f) ^(M) =T _(f) ^(M)−(T _(f) ^(L))^(M).  (14)Having this correction, one may calculate the phase by,φ=[(Toa ^(M) +ΔT _(f) ^(M)+Δ_(i) ^(M))%(1/f)]2πf  (15)where we use the “%” notation for the modulo or modulus operator, andToa^(M) is the measured time of arrival, as measured by M.

The acquisition sequence of each sub has to be synchronized to avoid anypotential conflict. All subs have to synchronize with the master clock.A local clock may drift slightly relative to the master clock, however,it should be readjusted once the drift exceeds a prescribed threshold.Every timing signal provides the actual clock difference between themaster clock and the local clock. Let T_(i) ^(L) and T_(i) ^(M) be theTOA of a timing signal, i, in local clock and master clock,respectively. We can establish a “tie” point as shown in FIG. 6. At theparticular instant of a tie point, the local clock and the master clockare synchronized. After power up, the first timing signal exchange leadsto the measurement of T_(tie) ^(M) and T_(tie) ^(L), and adjusting thelocal clock by (T_(tie) ^(M)−T_(tie) ^(L)) establishes the first tiepoint, Tie_(—)0. Note the clock adjustment may not be an actual hardwareadjustment, but rather can be a correction factor that is computed andused in subsequent calculations.

Using the tie point, we can calculate the difference between the masterclock and the local clock for any subsequent timing signal byreferencing to the tie point,Δ_(i)=(T _(i) ^(M) −T _(tie) ^(M))−(T _(i) ^(L) −T _(tie) ^(L)).  (16)For small values of Δ_(i), no correction is needed, but as these clockdiscrepancies increase, a new clock correction is needed. Whether acorrection is needed is decided by comparing Δ_(i) with a preset valueΔ_(adj) _(—) _(threshold), where the latter is decided by the user basedon the frequency of operation and the width of the excitation andreceiver windows. This parameter should be long enough to minimize thenumber of clock adjustments, yet short enough that the signal fallswithin the receiver acquisition window. Let Δ_(adj) _(—) _(i) be therequired local clock adjustment at each timing signal exchange. Then,

$\begin{matrix}\left\{ \begin{matrix}{if} & {{\Delta_{i} < {- \Delta_{adj\_ threshold}}},{\Delta_{adj\_ i} = {- \Delta_{adj\_ threshold}}}} \\{elseif} & {{\Delta_{i} > \Delta_{adj\_ threshold}},{\Delta_{adj\_ i} = \Delta_{adj\_ threshold}}} \\{{otherwise},} & {\Delta_{adj\_ i} = 0.}\end{matrix} \right. & (17)\end{matrix}$

After a clock adjustment is made, we have a new tie point. Let D_(i) bethe actual amount of adjustment needed by a local clock to stay in syncwith the master clock at each timing signal exchange. D_(i) iscalculated by,D _(i) =D _(i−1)+Δ_(adj) _(—) _(i)if D_(i)≧0, D_(i)=D_(i)% TCT_(length)if D _(i)<0, D _(i) =TCT _(length)−(−D _(i)% TCT_(length))  (18)where TCT_(length) is the duration of one complete TCT (tool controltable) acquisition cycle. Then, we can calculate the next acquisitionsequence starting time by,T _(si) ^(L)=(T _(i) ^(L) +D _(i)+TCT_(length))−(T _(i) ^(L) +D_(i)+TCT_(length)) % TCT_(length) −D _(i).  (19)

In normal operations, the transmitter pulses are sent as a sequence witha preset pulse width T (typically 10 ms). The receiver uses a longerreceiver sampling window, T+ΔT, (typically 20 ms), to capture the signalas it arrives at the receiver antenna. The measurements are performed bytransmitting a series of sequential, multi-frequency, single-tonepulses. For each pulse received, the pulse amplitude and time of arrival(TOA) are determined using a matched filter technique. The time ofarrival with respect to an arbitrary time reference in the receiver isconverted to a phase measurement. The amplitude and phase can be furtherprocessed.

In ideal conditions (i.e., when the clocks in the two modules areperfectly synchronized), the receiver sampling window is centered overthe received signal. In practice, with transmitter/receiversynchronization to within Δ_(adj) _(—threshold) , each received 10 mspulse is acquired through a 20 ms window, and is over-sampled beforebeing transferred to memory. By over-sampling, the receiver analogelectronics is kept to a minimum (pre-amplifying section plus a low passanti-aliasing filter), and the measurement processing is then fullydigital, allowing flexibility in algorithm development andimplementation. FIG. 7 shows a simulated received pulse after it hasbeen transmitted through a formation. This is a pure tone carriermodulated by a square window. Other pulses with wider bandwidth can bealso used, such as a pure tone convolved with a pseudorandom number (PN)sequence in order to be more resilient to coherent noise.

When the pulse of FIG. 7 is match filtered, the signal of FIG. 8 isobtained. As FIG. 8 shows, noise measurement may be performedimmediately after pulse acquisition. The signal of FIG. 8 is the resultof coarse processing, which is described below. Once the pulse waveformis acquired, cross-correlations with sliding 10 ms sine and cosinereference arrays are applied to the received signal to determine theamplitude and TOA to within the sampling time accuracy (coarseprocessing). Application of a matched filter provides the bestsignal-to-noise ratio (SNR). The processing diagram with the formulasused is shown in FIG. 9. The result from processing a simulated squarewindow 100 kHz pulse is a typical demodulation triangle. For coarseprocessing, nominal carrier frequencies are preferably chosen so thatthe reference cosine and sine functions can be stored in tables ofminimum size. Another broadband synchronization pulse would give asharper cross-correlation and could enhance the coarse estimation.

Once a coarse estimation has been made, an accurate determination of theTOA and amplitude may be performed using an accurate measurement of theclock discrepancy factor. The processing is similar to the coarseprocessing except that now a time-reversed signal corresponding to amatched filter is moved at the carrier frequency as seen by thereceiver. The received signal will be seen as a signal with a shiftedcarrier frequency, as is evident from the formula below (correspondingto a square window modulated sine wave), with Φ taken as the offset ofthe pulse waveform within one sampling time difference,

$\begin{matrix}{x_{i} = {{A \cdot {\sin\left( {{2 \cdot \pi \cdot \frac{f_{c}}{f_{s}} \cdot \left( {1 + {K_{ppm} \cdot 10^{- 6}}} \right) \cdot k} + \Phi} \right)}} + {N_{i}.}}} & (20)\end{matrix}$The fine TOA is estimated by first applying a Blackman (tapered) windowaround the received signal to remove the edge effect. The processingwill then filter the synchronization signal with the analyticrepresentation of the impulse response of the matched filter. For thecase of the square window modulated with a sine wave, that will be asine and cosine. The optimum location of the peak maximum is estimatedto within one sampling time. At the maximum, the optimum TOA withaccuracy better than the sampling time is obtained using,

$\begin{matrix}{\mspace{79mu}{{TOA}_{fine} = {\left( \frac{\Phi}{2\pi} \right) \cdot \frac{1}{f_{C} \cdot \left( {1 + {K_{ppm} \cdot 10^{- 6}}} \right)}}}} & (21) \\{\tan\left( {(\Phi) = {{\frac{\sum\limits_{k = 1}^{N}\;{x_{k} \cdot {\cos\left( {\omega \cdot k} \right)}}}{\sum\limits_{k = 1}^{N}\;{x_{k} \cdot {\sin\left( {\omega \cdot k} \right)}}}\mspace{14mu}{and}\mspace{14mu}\omega} = {2 \cdot \pi \cdot \frac{f_{c}}{f_{s}} \cdot {\left( {1 + {K_{ppm} \cdot 10^{- 6}}} \right).}}}} \right.} & (22)\end{matrix}$

FIG. 10 shows steps in an exemplary embodiment to determine the phase ofa signal when transmitter and receiver circuits use separate clocks. Thesteps comprise determining a drift factor between a master clock,measuring relative to a master clock reference frame, and a local clock,measuring relative to a local clock reference frame (step 500);determining an expected start time of a pulse sequence by the masterclock, relative to the master clock reference frame, and by the localclock, relative to the local clock reference frame (step 502);transforming the expected start time of the pulse sequence determinedrelative to the local clock reference frame to the master clockreference frame using the drift factor (step 504); determining adiscrepancy between the master clock and the local clock using theexpected start time of the pulse sequence determined by the master clockand the transformed expected start time of the pulse sequence from thelocal clock (step 506); determining a start time of a pulse within thepulse sequence by the master clock, relative to the master clockreference frame, and by the local clock, relative to the local clockreference frame (step 508); transforming the start time of the pulsedetermined relative to the local clock reference frame to the masterclock reference frame using the drift factor (step 510); determining acorrection factor between the master clock and the local clock using thestart time of the pulse determined by the master clock and thetransformed start time of the pulse from the local clock (step 512); anddetermining the phase using an expected time of arrival of the signalrelative to the master clock reference frame, the determineddiscrepancy, and the determined correction factor (step 514).

FIG. 11 shows a local terminal 605 equipped with a printer 610, a meansof capturing physical quantities 635, and a means of access 615 to anetwork 620 to which a server 625 is connected. The server 625 may befurnished with a database 630.

The local terminal 605 is, for example, a commonly used computer. Themeans of access 615 to the network 620 is, for example, a modem of aknown type permitting access to the network 620, for example theinternet. The server 625 is of a known type. The terminal 605 containssoftware that, when run, implements the steps in the process accordingto this disclosure. Alternatively, the terminal 605 does not containspecific software but implements a web browser and a web servicecontained in the server 625.

The terminal 605 may comprise a microprocessor 640 and memory unit 645containing an operating system 650 and application software 655containing instructions to implement the process according to thisdisclosure. Further, in a known manner, the local terminal 605 isequipped with a display screen 660 and means of control 665, forexample, a keyboard and a mouse.

The use of two or more tools within a bottomhole assembly may require acoordination of operations to either prevent or assure simultaneousoperations. For example, it may be desired that one tool be operatedsome desired time after the onset or completion of operations by anothertool. That is, a function onset time (meaning, broadly, some particularphase of the operation, e.g., onset, end, peak transmission, etc.) forone tool may be specified relative to a function onset time of anothertool. Other delay schemes are also possible. For simultaneousoperations, the delay is set to zero or the function onset times areequal. Getting the tools to operate with the desired temporal spacingcan be problematic if the tools use separate clocks. However, bydetermining the discrepancy and the correction factor between theseparate clocks, as described above, the desired function onset time canbe adjusted using the determined discrepancy and the determinedcorrection factor. The adjusted function onset time can then be used tocoordinate the operations of the tools.

As stated above, it is desired to synchronize slave clocks to areference master clock with minimum communication overhead betweensystems using these clocks. One could then preferably coordinate theaction and/or response of several downhole devices. One could also usesuch synchronized clocks in distributed measurement systems, in whichsome devices generate an excitation signal, and others receive andprocess the transmitted signal. Distributing the parts of a measurementsystem allows for greater transmitter-receiver spacing, which is usefulfor increasing the depth of investigation.

FIG. 12 illustrates one embodiment of a system, referred to herein as adisciplined clock system 200, constructed in accordance with thisdisclosure. The disciplined clock system 200 comprises an updatesubsystem 202 and a synthesis subsystem 204. The update subsystem 202operates at an update rate, based on the master clock (MC) phaseestimates, while the synthesis system 204 operates at a higher rate, thesynthesis rate. The disciplined clock system 200 may also include,either as an integrated or separate component, an MC clock phaseestimator 206 (represented as an input parameter in FIG. 12).

Update subsystem 202 comprises a phase-locked loop (PLL) device 208 anda parameter estimator 210. Update subsystem 202 tracks and providesupdated estimates of both the phase and frequency offsets each time itreceives an update from the MC phase estimator 206. Synthesis subsystem204 comprises a clock synthesizer 209 that produces a clock signal basedon those phase and frequency estimates.

For purposes of illustration and ease of discussion, we focus herein onthe problem of how to synthesize a disciplined clock system. Considertwo clocks: a master or reference clock (MC), and a slave clock (SC).Assume those two clocks are located on two different devices, a masterclock device (MCD) and a slave control device SCD, respectively. The SCDreceives updates from the MCD at certain intervals, and attempts tocontrol the slave clock so as to be as close as possible to the masterclock. From the perspective of the SCD, the master clock is a noisyclock that needs to be tracked.

One possible approach to clock synchronization is to sample the MCDclock phase and broadcast it to the SCD at the clock frequency orhigher. However, such an approach requires significant communicationoverhead. Therefore, preferably, clock information is sent at a ratemuch slower than the clock frequency. In an existing tool, referred toherein as a basis for comparison, the clock frequency is 12 MHz and ithas been chosen that the rate at which the MCD transmits timeinformation to the SCD(s) is once every 5.5 seconds. This period isknown to both the MCD and the SCD, hence there is no need to send anexplicit time stamp.

An exemplary operation scheme for the disciplined clock system 200 isillustrated in FIGS. 13A and 13B. The open circles (“dots”) representtimes when the MCD sends a signal to the SCD such that it can estimatethe MCD clock phase. If the clocks are perfectly synchronized to eachother, then from the perspective of the SCD, the clock phase of the MCDlooks like the “ideal clock” line. However, due to mismatch, itgenerally would look like the dashed line connecting the dots. In thatcase, the SCD must estimate and track this dashed-line process. Thearrows between the dots in FIG. 13A represent the output of thesynthesis subsystem 204, based on the phase and frequency estimates ateach update time. In FIG. 13B, each of the solid horizontal lines fromthe ordinate to a dot represents a time epoch of the MCD, while thedashed horizontal lines from the ordinate to a dot represents a timeepoch of the SCD. The arrows in FIG. 13B show the timing error betweenthe two epochs.

Using the timing scheme or sequence of the referenced existing tool inthe present method, and referring to FIG. 14, the MCD sends a timingsignal (square block) every 5.5 seconds based on its clock (dashed lines212). The SCDs correspondingly expect certain time(s) of arrival (TOA)of these signals according to their own clocks (solid lines 214). Theactual or measured TOA are shown by the dashed lines 216. The SCDssubtract the measured TOA from their own clock to obtain the clockerror, shown by the solid arrows 218. In the example shown, the clock ofthe SCD runs a little slower than the MCD clock, hence from theperspective of the SCD, it appears that timing signals arrive earlier astime progresses. Because the MCD communicates with the SCD only onceevery 5.5 seconds, the SCD can only update its estimate of the MCD clockevery 5.5 seconds. Hence we refer to 1/5.5 seconds as the update rate.An example of a distributed measurement system having one transmitterand one receiver is shown schematically in FIG. 15.

An example of the phase output of a “noisy” clock is shown in FIG. 16.The terms phase and time are used interchangeably herein. This ispermissible assuming the clock mismatch is sufficiently small that thereis no phase ambiguity. While the spectrum of an ideal clock approaches aDirac delta function in the frequency domain, the spectrum of a noisyclock is spread around the central frequency. A typical model for anoisy clock is based on a wide-sense stationary Wiener process and is astandard method taught in most communication textbooks. In most casesthere two primary factors: drift (frequency mismatch) and jitter. Thelatter is modeled as a white Gaussian process.

One possible approach for master clock phase estimation is for the MCDto periodically transmit a signal to the SCD, based on the MCD clock.The SCD can then estimate the phase/time of arrival of the receivedsignal to infer the phase of the MCD clock.

There is a class of methods from which one can obtain time of arrivalestimates at resolutions higher than the sampling rate. They are called“super-resolution” methods. There are two classes of super-resolutionmethods: those based on time-domain processing, and those based onfrequency-domain processing.

Time-domain based processing typically uses a fractional delay filter tocompute fractional delay hypothesis values. These typically require thatthe receiver have good prior information on what the fractional delayis, else it has to do a search. For each hypothesis value, the estimatormust compute the cross-correlation between the received signal and a(fractionally) delayed version of the pulse shape. Hence, in order toincrease the resolution by 10 times, the receiver has to construct 10correlation filters.

In contrast, frequency-domain based methods take advantage of a basictime-frequency duality. In the continuous-time domain:x(t)

X(ω)x(t−τ)

X(ω)exp(−jωτ)  (0.1)

In this approach, the time delay estimation problem is converted into amodel-based sinusoidal parameter estimation, and an autoregressivemethod is used to solve the problem. This technique can be refined byapplying newer line spectrum estimation algorithms with betterperformance.

In the sampled domain, one algorithm that can be used is known as ESPRIT(Estimation of Subspace Parameters via Rotational Invariance Technique).Suppose one observes a sampled version of the delayed noisy signaly[n]={x(t−τ)+z(t)}|_(t=n/fs) at the SCD, sampled at rate f_(s). Assumingthat there is no frequency-domain aliasing, the discrete time Fouriertransform (DTFT) is given by:Y(e ^(jΩ))=Y(ω)|_(ω=Ωifs), Ω∈(−π,+π)  (0.2)

Further, assuming that there is no spatial aliasing, the discreteFourier transform (DFT) is given by:Y[k]=Y(e ^(jΩ))|_(Ω=2πk/N) =Y(ω)|_(ω=2πkfs/N) ,k=0, . . . ,N−1  (0.3)Hence,Y[k]=X[k]exp(−j2π

k

fs

τ/N)+Z[k]  (0.4)

In the above, X [k] is the DFT of the sampled version of x(t), and Z[k]represents the noise. Thus, we have mapped the desired time delayparameter τ into a complex exponential parameter.

For our problem, the algorithm can be explained very simply. Let thecross-correlation between the received signal y[n] and the sampledversion of template signal x[n] be given by:

$\begin{matrix}\left. \begin{matrix}{{r\lbrack n\rbrack} = {{x\left\lbrack {- n} \right\rbrack}*{y\lbrack n\rbrack}}} \\{= {{x\left\lbrack {- n} \right\rbrack}*\left( {{x\left\lbrack {n - \tau} \right\rbrack} + {z\lbrack n\rbrack}} \right)}}\end{matrix}\Leftrightarrow\begin{matrix}{{R\lbrack k\rbrack} = {{X^{*}\lbrack k\rbrack}{Y\lbrack k\rbrack}}} \\{= {{X^{*}\lbrack k\rbrack}\left( {{{X\lbrack k\rbrack}{\exp\left( {{- {j2\pi}}k{fs}{\tau/N}} \right)}} + {Z\lbrack k\rbrack}} \right)}} \\{= {{{{X\lbrack k\rbrack}}^{2}{\exp\left( {{- {j2\pi}}k{fs}{\tau/N}} \right)}} + {{X^{*}\lbrack k\rbrack}{Z\lbrack k\rbrack}}}}\end{matrix} \right. & (0.5)\end{matrix}$

From Equations (0.4) and (0.5), we can see that in discrete-time wehave: R[k]/R[k−1]∝exp(−j2π

fs

τ/N) for all k. Thus a reasonable strategy for estimating τ is tocompute point-wise ratios of R[k], R[k−1] to infer the quantity exp(−j2π

fs

τ/N), and to average the estimate. To estimate this efficiently, we formtwo vectors R₁ and R₂ for the first N−1 and last N−1 samples of X[k],respectively. Then we compute the right inverse of R₁ from R₂ to obtainan estimate of exp(−j2π

fs

τ/N). In the presence of noise, we operate on the cross-correlationinstead of the direct measurement.

In summary, the algorithm is given by: (1) computing, preferablyoffline, the N-point DFT of the transmitted signal x[n]; (2) receivingy[n] and computing the cross-correlation r[n]=y[n]*x[−n]; (3) findingthe peak, n*, of the magnitude of the cross-correlation |r[n]| (this isthe integer sample delay estimate); (4) computing the DFT of r[n], n=n*,. . . ,n*+N−1, say R[k]; (5) forming R₁={R[0], . . . ,R[N−2]}, R₂={R[1],. . . , R[N−1]}; (6) computing u=R₂ \R₁ (rotational invariance step);and (7) computing

${= {\left( {n^{*} - {{{angle}(u)}\frac{N}{2\pi}}} \right) \times \frac{1}{fs}}},$where f_(s) is the sampling rate of the analog-to-digital converter(ADC).

It is important to note that the resolution of the computation of τ doesnot depend on the sampling rate f_(s). The performance of this system isbetter with increasing SNR, independent of the sampling rate f_(s). Itis straightforward to extend this algorithm to include multiple arrivaltimes. Because this frequency-domain method can resolve pulses witharrival time spacing smaller than the sampling rate, it is considered asuper-resolution method.

For correlation-based TOA estimation, it is known that betterperformance is possible with higher signal bandwidth. The Cramer-Raobound on time of arrival estimation is well-studied, and is valid formedium-to-high SNR. Suppose that the transmitted signal x(t) has energyE and is corrupted with additive white Gaussian noise (AWGN) having aspectral density of N₀/2. The bound is given by:

$\begin{matrix}{{{{var}{()}} \geq \frac{1}{\frac{E}{N_{0}/2}F^{2}}},} & (0.6)\end{matrix}$where F² is the mean-square bandwidth of the signal. Hence, for the samesignal-to-noise ratio, wider mean-square bandwidth means betterperformance.

However, the Cramer-Rao bound is not valid for low SNR cases, or forsignals with poor autocorrelation. One example is a signal having manyfalse correlation peaks. One could then use, for example, the Ziv-Zakaibound, which considers the probability of mistaking the correct peak foran incorrect one.

One possible choice that gives good TOA performance is a spread spectrumsignal, which is a sequence of pulses modulated using a pseudorandomsequence. It has the advantage of having both a narrow autocorrelationwith no ambiguous peaks (and by duality, a large bandwidth) and a smallpeak-to-average ratio.

${x(t)} = {\sum\limits_{\ell = 0}^{L - 1}\;{a_{\ell}{p\left( {t - {\ell T}} \right)}}}$We choose the sequence {a_(l)} to be a pseudorandom sequence such as anM-sequence. The rate R=1/T is called the symbol or chip rate. Togetherwith the pulse shape, it determines the bandwidth of the signal.

The phase-locked loop (PLL) 208 (see FIG. 17) preferably is able tocontend with both drift and jitter. Thus, it is preferably at least asecond-order PLL. Design and analysis of a PLL is standard in mostcommunication textbooks. A second order PLL is described as follows. Letx[n] be the input sequence. The loop filter can be chosen to have thefollowing transfer function:

$\begin{matrix}{{y\lbrack n\rbrack} = {{K_{p}{x\lbrack n\rbrack}} + {\sum\limits_{m = {- \infty}}^{n}\;{K_{i}{x\lbrack m\rbrack}}}}} & (0.7)\end{matrix}$Since the phase estimate {umlaut over (x)}[n] is the output of anintegrator that takes y[n] as its input, then y[n] is the instantaneousfrequency offset estimate. The parameters K_(p) and K_(i) can be chosenaccording to the desired damping factor and the expected frequencyoffset that is to be tracked. Those parameters can be tuned to trade offbetween tracking speed, damping factor, and variance at the output.

Going back to the system shown in FIG. 14, the timing error is chosen tobe the input sequence to the PLL 208. Then {umlaut over (x)}[n] will bean estimate of the timing deviation and y[n] will be an estimate of thefrequency deviation, normalized to the update rate (since its samplingperiod is 5.5 seconds).

In summary, two outputs from the PLL 208 are obtained: (1) the phaseestimate {umlaut over (x)}[n]; and (2) the frequency estimate y[n].These estimates are updated once per reception of timing signal from theMCD, and the updated estimates are used to synthesize a clock signal atthe SCD.

The purpose of the parameter estimator is to obtain a reliable estimateof both the frequency and phase offsets that can be used to synthesize aslave clock signal to maintain tracking until the next update instance.From the output of the PLL 208, one obtains a sequence of phaseestimates after filtering. The derivative of this sequence (which is theoutput of the loop filter) gives a frequency offset estimate.Alternatively, as mentioned before, the frequency estimate can bedirectly obtained from the output of the loop filter prior to passing itinto the integrator. Based on estimates of frequency and phase offsets,the clock synthesizer 209 will add corrections to the local slave clockcontinuously or at the synthesis rate.

One possible approach for clock phase estimation is for the MCD toperiodically transmit a signal to the SCD, based on the MC. The SCD canthen estimate the phase/time of arrival of the received signal to inferthe phase/time of the MC. Based on basic estimation theory, the signalmay be chosen to be a spread spectrum signal, as described above:

${x(t)} = {\sum\limits_{k = 0}^{K - 1}\;{a_{k}{p\left( {t - {kT}} \right)}}}$Again, the sequence a_(k) is chosen to be a pseudorandom sequence. Fordefiniteness, we limit the span of the overall signal to 20 ms, thoughthis is not meant to be a limitation on the method. K and T are chosenaccordingly, and as before, the rate R=1/T, is the symbol or chip rateand it, along with the pulse width, determines the bandwidth of thesignal.

Simulation results of the super-resolution method based on ESPRIT arepresented. The system makes a coarse estimate based on correlationpeak-finding and then uses ESPRIT on a windowed signal around the peak.Three pulse shapes (see FIG. 18) are compared, and ordered in FIG. 18according to their bandwidths: (1) a root raised-cosine pulse (RMSbandwidth at R=10 kHz is 18.7 kHz); (2) a square pulse (RMS bandwidth atR=10 kHz is 97 kHz); and (3) a biphase pulse (RMS bandwidth at R=10 kHzis 181 kHz). The latter two are two-level pulses that can be generatedusing a simple 1-bit DAC. From FIG. 19, it can be seen that theperformance scaling law predicted by Equation (0.6) is achieved. Goingfrom R=10 kHz to R=20 kHz gives a 3 dB improvement, and going from R=10kHz to R=50 kHz gives a 15 dB improvement. Further, pulses with highertime-bandwidth products have better performance than those with lowertime-bandwidth products. As used herein, the per-sample SNR is definedto be the ratio between the sampled signal energy and the per-sampleAWGN variance, as given by the sampling rate.

Note that the biphase pulse is nothing but a square pulse with a signchange within it. Further inspection reveals that at high SNR, theperformance of a square pulse at a 20 kHz pulse rate is nearly identicalto that of a biphase pulse at 10 kHz. Further, it should be noted thatin all cases, the performance of the instant method is within 2 or 3times the bound of Equation (0.6), which is re-computed for each choiceof pulse shape and symbol rate R.

Although the above discussion focuses on baseband pulses, it is easy toconvert the system to its passband equivalent. Further, moving to thepassband representation will improve the performance as predicted inequation (0.6), since the mean-square bandwidth of the signal will beimproved.

Recalling again the timing scheme of the referenced existing tool, themaster and slave clock frequencies are 12 MHz, with accuracy guaranteedup to 40 ppm. That means that the maximum offset is 480 Hz. The updaterate is, again, once every 5.5 seconds. The performance analysis on theupdate subsystem 202 of the present disclosure is easy to carry to thesynthesis subsystem 204 since its behavior is completely determined bythe update subsystem 202. Exemplary error outputs are shown in FIGS. 20and 21, using the timing scheme of the previously referenced existingtool. As stated before, parameters can be tuned to trade off betweentracking speed, damping factor, and variance at the output, though thatwas not done in those examples. The case shown is the absolute worstcase. It assumes that the clocks are immediately mismatched by 480 Hz.In reality, clocks are very accurate at room temperature and theirmismatch drifts slowly as temperature increases.

The above analyses assume that we are able to obtain accurate estimatesof the MC phase. In reality, that is not generally the case. One may belimited by several factors, including jitter, the effect on the signalthat relays phase of noise on the communication bus, and the front-endparameters of the SCD (e.g., sampling rate, noise floor, etc.). However,one can still do communication-style link budget analysis.

Assume that the jitter in phase estimation is the sum of jitter due tothe MC jitter and the estimation error due to noise afflicting thesignal relaying the phase, sent from the MCD to the SCD. To estimatetime of arrival, one should consider the time of arrival estimation inthe presence of AWGN. Assume that the noise density on the bus islimited to 200 μV/RtHz. For the timing signal, we assume a worst-case of1 Vpp (peak-to-peak) with a duration of 20 ms. Hence, the SNR isapproximately 46 dB.

Based on the bound of Equation (0.6), we can derive that, in this case,it is possible to obtain a root-mean-square (RMS) time delay estimationerror as low as 700 ns. However, this bound is derived for acontinuous-time estimator and ignores the issue of sampling rate.Further, the transfer function of the propagation channel itself hasbeen ignored.

In the present implementation, the sampling rate of the SCD timingsignal receiver is 500 kS/s. This means that the sampling interval is 2μs. Of course, the ADC itself has some jitter. Since the samplinginterval value is already very close to the RMS of the time of arrivalestimation error, it is likely that an increase in sampling rate wouldbe beneficial. This also means that if the signal shape is furtheroptimized (in terms of mean-square bandwidth), then the sampling rate ofthe system is preferably increased so that it does not become thebottleneck.

Simulation results using the referenced prior art method, i.e., based oncarrier phase, are compared with simulation results from an embodimentof the present method, and shown in FIG. 22. The prior art method sendsa single tone signal at 100 kHz, and requires a high SNR to get goodresults. At lower SNR values, the higher choice of carrier frequency f₀gives the worst performance due to the aliasing effect from carrierphase ambiguity. The cutoff SNR below which phase ambiguity plays a roleis at approximately 50 dB for f₀=150 kHz and at approximately 40 dB forf₀=50 kHz or 100 kHz. By comparison, the new signaling choices do notsuffer the phase ambiguity problem until one approaches the 20-30 dBregime. This means the new signaling choices are more robust to noise,as well as being naturally more robust to the single-tone interferencethat, due to motor noise, is prevalent in the BHA. Further, improvingthe prior art peak-finding method using the carrier phase only hasmarginal improvement beyond the sampling period of the ADC, as shown bythe horizontal line near the center of FIG. 22.

By comparison, the results from the embodiment of the present method ismonotonically better with higher bandwidth or rate. That means that ifone requires higher performance, there are parameters that one canexploit to improve the performance. Although the RMS bandwidth of thesignal of the embodiment of the present method is lower than that usedby the prior art method, this can be remedied by choosingpassband-modulated equivalents with minimal change to the estimationalgorithm.

Particular deficiencies of the prior art method include: (1) even thoughthe RMS bandwidth of the signal is larger than that of the embodiment ofthe present method, the poor choice of signal means that the prior artmethod suffers from phase ambiguity; and (2) the RMS bandwidth of thesignal is also wasted due to processing mostly in the time domain.Because the present estimation technique is independent of samplingrate, it can be applied to any signal, such as the signal of the priorart method. However, the estimation will still suffer from phaseambiguity.

To better illustrate, another comparison is shown in FIG. 23, in whichthe RMS errors for the present (i.e., “new”) and prior (i.e., “old”)techniques are plotted versus different RMS bandwidth of differentsignals. These are also compared with the Cramer-Rao bound (downwardlysloping line). Clearly, the results using an embodiment of the presentmethod are shown to be close to the bound, whereas the old (i.e., priorart) method and associated signals are all very far from the bound.

To further refine the measurement, one can add the time of arrivalestimation error and the phase noise from the MCD. Based on the clockspecification, which in our example is 40 ppm at 12 MHz, the timeuncertainty from one transmission of the timing signal to the next (5.5s) is 40 ppm×5.5 s=220 μs. The question is then whether most of this isjitter or drift. For now, we assume that the jitter from the MC is lowerthan the jitter from the TOA.

Having derived the jitter in phase estimation, we now investigate theeffect on the phase and frequency estimate by linearized analysis of thePLL. We know that the loop bandwidth of the PLL has to be at least wideenough to track the frequency offset (in this case, it is 480 Hz). Thereferenced prior art method computes the difference between phaseestimates to obtain a frequency estimate, followed by a median filter oflength 5. The median filter is nonlinear, hence resort is made to simplesimulation to evaluate the performance of phase tracking. This is shownin FIG. 24. Even if the old TOA method is used, the performance isapproximately 30 dB better with the PLL. This is not surprisingconsidering the prior art method only does differential estimation.However, if we switch to the new TOA method, then we obtain even moreperformance gain. As shown, the frequency mismatch can be reduced fromjust below 1 μHz to below 10 nHz. This is an order of magnitudeimprovement in the clock mismatch RMS.

Thus, the present disclosure has disclosed a system and method for clocksynchronization between a Master Clock Device (MCD) and one or severalSlave Clock Devices (SCDs), where the goal is for the SCDs to be able toproduce the clock of the MCD with high accuracy but with minimalcommunication overhead. The disclosed method can include one or more ofthe following aspects: (1) a new timing signal; (2) a new time ofarrival estimation technique; and (3) a new clock tracking technique. Itwas shown that careful choice of the timing signal can improve theperformance and consistency of a time propagation system. The new timeof arrival estimation technique is an improvement over prior artmethods, and works for any choice of timing signal, though it will notcure the poor correlation properties and phase ambiguities of the “old”timing signals. Finally, the present clock tracking technique issuperior to prior art techniques, regardless of how the timing signaland/or the time of arrival estimation is done. FIG. 24 shows that, for aparticular embodiment of the present method, the RMS error of frequencyestimation can be lowered by 100 times compared to a particular priorart system.

The method described in this document can be used to synchronize severaldevices together. This includes multiple subs in the same BHA, ordevices across several shuttles, or generally devices in severallocations where there is limited communication between them. The timingsignaling scheme can also be changed from sending a signal at regular,pre-defined intervals to sending signals occasionally, but with a timestamp. In that case, the PLL is preferably changed from a regular PLL toa gated PLL, and the analysis is slightly modified accordingly.

As stated above, when measuring downhole formation characteristics,sometimes transmitters and receivers are widely separated in a toolstring. Signals are often sent into the formation by one or moretransmitters, and then received by one or more receivers at particulardistances from the transmitter(s). The received signal is analyzed toattempt to infer formation characteristics.

One important measurement is the time from when a transmitter is fireduntil that signal is received. Measuring the delay between a transmitterfiring and the reception of the signal at a receiver allows the signalvelocity through the formation to be measured, from which one can inferformation characteristics. In some cases, it is preferable for thisdelay to be measured to an accuracy of better than 30 ns.

Further reiterating, when transmitters and receivers are not too widelyspaced, it is possible for the receivers to know the time of thetransmitter firing through many conventional mechanisms. One suchmechanism is for the same oscillator signal to be used for thetransmitters and receivers, thus allowing the clocks to remainsynchronized. Another mechanism is each transmitter can send a signalthrough a dedicated wire to notify receivers that they have fired.However, with longer spacings between transmitters and receivers, thoseconventional approaches may not work. It may not be possible to run adedicated wire through this distance, nor can the same oscillator signalbe made available to all transmitters and receivers.

One approach to overcome these limitations is to periodically send asine wave as a synchronization signal through a common tool bus. Forexample, a 10 millisecond burst of a 100 KHz signal might be sent every5 seconds as a synchronization signal. However, this approach has somedifficulties. One problem is that sine waves can correlate every ±2πradians, with very little difference between the summation of the propercorrelation and the ±2π offset correlation. For the example above, thisdifference is only about 0.01%, so fairly small amounts of noise canlead to an offset correlation. When an offset correlation occurs, thisleads to a timing error of 10 μs, far in excess of the 30 ns accuracydesired. Another problem is that sine waves are strongly affected bytone noise near the frequency used. When such noise occurs, the signalcan fail to correlate. What is desired is robust timing that stronglyresists miscorrelation, and is not subject to failure from tone noise.

Instead of transmitting a single tone as a timing signal, one can send awideband timing signal with strong correlation properties. The strongcorrelation properties can be ensured by using a baseband binary codethat forms a maximum length pseudorandom (PN) sequence. The widebandsignal is received and correlated with the reference transmitted signalto obtain an accurate arrival time. Such robust and accuratesynchronization allows for extremely accurate recovery of timinginformation, even with high levels of noise and with channels that havemultiple phase inversions. There is no point of possible miscorrelationwithin the PN timing signal and the wideband signal provides resistanceto tone noise at any frequency. That is, the coding gain from thecorrelation of the PN sequence provide effective signal amplification,allowing resistance to extremely high levels of Gaussian or tone noise.

Referring to FIG. 25, a maximal length pseudorandom (PN) sequencegenerator 610 produces a PN sequence of the desired length. A longersequence can provide more coding gain and better correlation, but italso takes more time to transmit. Typical lengths would be 512 to 4096bits, with longer or shorter lengths preferable in various situations.

This binary PN sequence is sent to a modulator 620. The modulator 620modulates the PN sequence to produce a modulated PN sequence. Typicalmodulation might include pulse width modulation (PWM), biphasemodulation, BPSK modulation, or a variety of other modulation techniqueswhich produce wideband signals. In many cases, the best modulationchoice may depend upon the characteristics of the transmitter 630 andtool bus 640. The modulator 620 may also pre-emphasize the signal tocompensate for the characteristics of the transmitter 630, so that thewideband reference signal transmitted though the tool bus 640 has a moreor less flat frequency spectrum. Additionally, the modulator 620 couldalso pre-emphasize the signal based on anticipated characteristics ofthe tool bus 640 channel.

The transmitter 630 couples the signal from the modulator 620 to thetool bus 640. One aspect of the transmitter 630 is that the signal isput on the tool bus 640 with very specific and highly accurate timing.By predefining when the signal will be coupled to the tool bus 640, areceiver 650 can know precisely when the local transmitter 630 put thesignal on the bus 640, so that synchronization can be more precise. Toolbus 640 interacts with the wideband timing signal coupled by thetransmitter 630, delivering a signal that is affected by thisinteraction to the receiver 650.

The receiver 650 is coupled to the tool bus 640 to retrieve the signalat the receiver 650. In the receiver 650, the signal is typicallyamplified, passed through anti-aliasing filters, and is sampled at arate that is at least twice as high as the maximum frequency in thetiming signal. The resulting received timing signal is the passed to thecorrelator 660, typically as an array of digital samples.

The correlator 660 typically has an array of digital samples stored thatcorresponds to the wideband timing signal generated by the transmitter630. This array is correlated with the received timing signal array fromthe receiver 650. Because of the strong correlation strength of the PNsequence, a good correlation can be made regardless of the channelcharacteristics of the tool bus 640. This correlation provides a precisearrival time for the timing signal. This arrival time may be used by theclock correction or clock compensation unit 670.

If clock correction unit 670 implements a clock correction, the localclock is adjusted using the arrival time and the predefined value forwhen the timing signal originated from the transmitter 630 to set thelocal receiver clock to a value corresponding to the estimated value ofthe transmitter clock. A second order timing loop can track frequencydrift, so that the clock frequency can be corrected between timingsynchronization points. If clock compensation unit 670 implements clockcompensation, the local clock is left free-running and is not adjusted.Instead, compensation factors are calculated which can be used with thelocal clock to calculate the estimated value of the transmitter clockwhenever this value is needed. Because the second order timing loop cantrack frequency drift, the clock compensation factors can be adjustedbetween timing synchronization points to reflect the frequencydifference. Regardless of whether clock compensation or clock correctionis used, the result is that the receiver 650 can accurately estimate thevalue of the transmitter clock at any time.

Accurate timing allows, among other things, the delay between widelyseparated tool measurements to be accurately determined. If atransmitter sends a signal into a formation at a known local time, whena receiver detects the signal, it can determine what the time on thetransmitter clock was at the time the signal was detected by thereceiver. By subtracting this time from the known time when thetransmitter originates the signal, the delay of the signal through theformation can be calculated. This allows widely separated toolmeasurements to still have accurate formation delay measurements.

A prototype apparatus has been built and the disclosed methodimplemented using a 1024 bit maximal length PN sequence using biphasemodulation at a 62.5 KHz symbol rate. This prototype implementation wastested in the laboratory. When Gaussian or tone noise that was 2.5 timeslarger than the signal was injected, timing was still recovered withless than 2 ns standard deviation. When the timing signal was testedwith a worst case tool bus, the timing was recovered with less than 2.2ns standard deviation. Thus, this apparatus and method has been provenin the laboratory to be accurate and robust.

Once again, a thesis of this disclosure is that many situations requiremeasuring the phase of a received signal very accurately. For example,using such phase measurement to determine the delay between transmissionand reception of a measurement signal allows one to determine thepropagation velocity of the channel being measured. The accuratelymeasured phase may also be used to synchronize clocks between varioussystems. In addition, there are many other situations where accuratephase measurement is needed.

As described above, a highly correlated signal such as a maximal lengthPN sequence is preferably used when accurate measurement of receivedphase is needed. For such a signal, if the approximate signal to bereceived is known in advance, a reference signal can be stored in thereceiver and correlated with the received signal. The point of maximumcorrelation corresponds to the phase of the received signal.

This point of correlation has a symbol phase resolution based on thesampling rate and the symbol rate. If the signal is sampled at eighttimes the symbol rate, then each sample corresponds to ⅛ of 360 degrees,or 45 degrees of the symbol rate. If a greater phase accuracy is needed,then the sampling rate per symbol could be increased. There are,however, often limitations restricting sampling rate, such as maximumspeed of an A/D converter. These limitations can restrict the maximumphase accuracy available from increasing the sampling rate. Othertechniques such as zero-crossing detection, curve fitting, etc. canresolve phase below the sampling rate. However, these techniques eitherhave accuracy limitations (especially when noise exists) or very highcomputational requirements. What is desired is a method to resolve thephase with a resolution at least 50 times greater than the samplingrate, and having fairly low computational requirements.

To that end, one may use a dual correlation (i.e., two correlations) ofa signal with strong correlation properties. The strong correlationproperties can be ensured by using a baseband binary code that forms amaximum length pseudorandom (PN) sequence, as described above.

As an illustrative embodiment, a first correlation is the typicalcorrelation of the received signal with a stored reference signal.However, instead of just using the point of highest correlation, nearbycorrelation points are also used. For example, three points on eitherside of the maximum point could be used, selecting a total of sevenpoints. The selected points near the maximum correlation are used for asecond correlation. They are correlated with an array of correlationpoints that correspond to various correlation phases. The length of thisarray could correspond to the amount of subsample resolution needed. Forexample, if a phase resolution 50 times greater than the sampling rateis needed, then an array of 50 correlation points corresponding to 50correlation subphases can be used. Alternatively, a smaller array can beused, with interpolation used to find the subphase between two tablepoints. This approach allows for lower computational cost compared tothose required for curve fitting. Extremely accurate phase recovery ispossible, and such accurate phase recovery may enable applications thatmight not otherwise be possible.

FIG. 26 shows a correlation of a PN sequence that was sampled at a rateof eight samples per symbol. Note the each phase has a unique curve,based on the center correlation values, and the plots are symmetricalaround the 22.5 degree phase point. The symmetry around the 22.5 degreepoint is based on the sampling rate of eight samples per symbol, so eachsample corresponds to a 45 degree phase. As the phase passes through the22.5 degree point, the maximum correlation value moves to the nextsample. At the 22.5 degree point, the correlation points aresymmetrical.

When correlating each received correlation midpoint value againstreference correlation midpoint values for each phase, the correlationwith the highest value corresponds to the closest received phase.Because the correlation phases are symmetric about the 22.5 degreepoint, it is only necessary to store correlation values between 0 and22.5 degrees. Correlation values between 22.5 degrees and 45 degrees areeasily found based on the symmetry.

An input signal (see FIG. 27) for which one wishes to determine thephase is preferably a signal with high correlation strength such as amaximal length PN sequence. In the receiver 710, the input signal istypically amplified, passed through anti-aliasing filters, and issampled at a rate that is at least twice as high as the maximumfrequency in the input signal. Preferably, a hardware analog high-passfilter is also used prior to the anti-aliasing filers to remove thelow-frequency noise and DC offset. The resulting received signal ispassed to a first correlation unit 720, typically as an array of digitalsamples. The first correlation unit 720 typically has an array ofdigital samples stored that corresponds to the expected received signal.Because of the strong correlation strength of the PN sequence, if a PNsequence is used as the input signal, then a good correlation can bemade even if the input signal is highly noisy or distorted. Signals withless correlation strength will be less resistant to noise anddistortion.

To minimize processing time, the first correlation can be done in twosteps. A coarse correlation can be done using only a subset of thestored array of digital samples that corresponds to the expectedreceived signal. Once the approximate correlation point is determined, afine correlation is performed to find the correlation midpoints for thefirst correlation. The first correlation unit 720 determines the pointof highest correlation and the nearby points, referred to herein as thecorrelation midpoints. Those correlation midpoints are passed to asecond correlation unit 730.

The second correlation unit 730 has an array of correlation referencepoints stored, with each row of the array corresponding to a particularphase. Depending upon the phase resolution desired and computation timeavailable, this array may typically have between ten and 100 rowscorresponding to various phases. The correlation midpoints arecorrelated with the stored array of correlation reference points. Thisconstitutes the second of the two correlations. The point of highestcorrelation indicates the row of the array corresponding to the bestphase estimate.

A curve fitting approach can be used for peak correlation detection whenhigher phase resolution is needed, but a larger array of correlationreference points is impractical, due to storage space, computationalpower, or other considerations. To obtain high phase resolution, thethree highest points of correlation for the second correlation are used.The corresponding phases are curve fitted and peak correlation detectionis performed to find the best phase estimate.

A prototype apparatus was built and the method implemented using dualcorrelation with an adjustable array of correlation points thatcorrespond to various correlation phases. Dual correlation using 10 to100 reference correlation points was tested. Such testing was done in alaboratory using input signals that were sent through bus conditionswith high signal distortion. In addition, Gaussian and tone noise wasadded to the input signals. Even with very high distortion and noise,the dual correlation process found the input phase very accurately, witha standard deviation of about 0.1 degrees.

To reduce the demodulation error caused by hardware effects, digitalcaptures of actual received signals may be used to generate thedemodulation reference vectors. By using the actual signals to generatethe reference vectors, the reference vectors can better match the actualhardware implementation.

It should be appreciated that while the invention has been describedwith respect to a limited number of embodiments, those skilled in theart, having benefit of this disclosure, will appreciate that otherembodiments can be devised which do not depart from the scope of theinvention as disclosed herein. Accordingly, the scope of the inventionshould be limited only by the attached claims.

What is claimed is:
 1. A method comprising: providing a disciplinedclock system comprising an update subsystem and a synthesis subsystem,wherein the update subsystem operates at an update rate and thesynthesis subsystem operates at a synthesis rate that is faster than theupdate rate; providing a first clock phase estimate to the updatesubsystem; determining a frequency offset estimate and a phase offsetestimate using the first clock phase estimate and the update subsystem,wherein determining the phase offset comprises estimating the time ofarrival of a signal; providing the frequency offset estimate and thephase offset estimate to the synthesis subsystem; and using thesynthesis subsystem to determine a clock signal based at least partiallyupon the frequency offset estimate and the phase offset estimate.
 2. Themethod of claim 1, wherein estimating the time of arrival for the signalcomprises using a super-resolution method.
 3. The method of claim 1,wherein the signal has a strong correlation property.
 4. The method ofclaim 1, wherein the signal comprises a wideband signal or apassband-equivalent of the wideband signal.
 5. The method of claim 1,wherein the signal comprises a spread spectrum signal, a chirp signal, apseudo-random signal, or a passband-equivalent of those signals.
 6. Themethod of claim 1, wherein determining the frequency offset estimate andthe phase offset estimate comprises using a phase-locked loop.
 7. Themethod of claim 1, further comprising tracking a mismatch between theseparate clocks.
 8. The method of claim 1, further comprisingreproducing a first clock at a second clock using the synthesissubsystem.
 9. The method of claim 1, further comprising compensating fora signal propagation time through a bottomhole assembly using the clocksignal.
 10. The method of claim 1, wherein the disciplined clock systemfurther comprises a master clock phase estimator.
 11. The method ofclaim 10, wherein the master clock phase estimator provides the firstclock phase estimate.
 12. The method of claim 1, wherein the signalcomprises a signal relative to a master clock reference frame.
 13. Themethod of claim 12, wherein determining the phase offset additionallycomprises using a correction factor and a discrepancy between a masterclock and a local clock.
 14. The method of claim 13, wherein thecorrection factor is determined between the master clock and the localclock using a start time of a pulse determined by the master clock and atransformed start time of a pulse from the local clock.
 15. The methodof claim 13, wherein the discrepancy is determined between the masterclock and the local clock using an expected start time of a pulsesequence determined from the master clock and a transformed start timeof a pulse sequence determined from the local clock.
 16. An apparatuscomprising: a disciplined clock system having: an update subsystem thatoperates at a first rate and is configured to receive a clock phaseestimate signal and output a frequency offset estimate and a phaseoffset estimate based on the received clock phase estimate signal,wherein the phase offset estimate is at least partially determined byestimating the time of arrival of a signal; and a synthesis subsystemthat operates at a second rate that is greater than the first rate andis configured to receive the frequency offset estimate and the phaseoffset estimate and output a clock signal determined based at leastpartially upon the received frequency offset estimate and the receivedphase offset estimate.
 17. The apparatus of claim 16, wherein the updatesubsystem comprises: a phase-locked loop that receives the clock phaseestimate signal; and parameter estimator logic that receives an outputof the phase-locked loop and provides the frequency offset estimate andthe phase offset estimate.
 18. The apparatus of claim 16, wherein thesynthesis subsystem comprises a clock synthesizer that receives thefrequency offset estimate and the phase offset estimate and provides theclock signal.
 19. The apparatus of claim 16, wherein the apparatuscomprises a downhole well logging tool.